Question

The molecular formula of aspartame, the artificial sweetener marketed as NutraSweet \({ }^{\infty}\), is \(\mathrm{C}_{14} \mathrm{H}_{18} \mathrm{~N}_{2} \mathrm{O}_{5}\). (a) What is the molar mass of aspartame? (b) How many moles of aspartame are present in \(1.00 \mathrm{mg}\) of aspartame? (c) How many molecules of aspartame are present in \(1.00 \mathrm{mg}\) of aspartame? (d) How many hydrogen atoms are present in \(1.00 \mathrm{mg}\) of aspartame?

Short answer

(a) The molar mass of aspartame is approximately 294.3 g/mol. (b) There are approximately \(3.40 \times 10^{-6}\) moles of aspartame in 1.00 mg. (c) There are approximately \(2.05 \times 10^{18}\) aspartame molecules in 1.00 mg. (d) There are approximately \(3.68 \times 10^{19}\) hydrogen atoms in 1.00 mg of aspartame.

Step-by-step solution

Calculate the molar mass of aspartame

To find the molar mass of aspartame, we need to multiply the number of each element in the molecule by their respective atomic masses and add them up. The atomic masses are approximately: - Carbon (C): 12.01 g/mol - Hydrogen (H): 1.01 g/mol - Nitrogen (N): 14.01 g/mol - Oxygen (O): 16.00 g/mol Molar mass of aspartame = (14 x 12.01) + (18 x 1.01) + (2 x 14.01) + (5 x 16.00)

Calculate the moles of aspartame in 1.00 mg

To find the moles of aspartame in 1.00 mg, we need to divide the mass by the molar mass. First, convert the mass from mg to g. 1.00 mg = 0.00100 g Moles of aspartame = mass / molar mass = 0.00100 g / molar mass (from step 1)

Calculate the number of molecules in 1.00 mg of aspartame

Now we need to multiply the moles of aspartame by Avogadro's constant (6.022 x 10^23) to find the number of molecules. Number of aspartame molecules = moles of aspartame (from step 2) x (6.022 x 10^23)

Calculate the number of hydrogen atoms in 1.00 mg of aspartame

Finally, we need to multiply the number of aspartame molecules by the number of hydrogen atoms in one aspartame molecule to find the total number of hydrogen atoms. Number of hydrogen atoms = number of aspartame molecules (from step 3) x 18

Key concepts

Molecular Formula

The molecular formula is a way to represent the number and type of atoms in a molecule. For aspartame, the molecular formula is \(\text{C}_{14}\text{H}_{18}\text{N}_2\text{O}_5\). This signifies that each molecule of aspartame consists of:

• 14 carbon atoms
• 18 hydrogen atoms
• 2 nitrogen atoms
• 5 oxygen atoms

The molecular formula gives us a detailed count of each element in the compound. This count is crucial for calculating the molar mass, predicting reactivity, and understanding the chemical properties of the compound. In essence, the molecular formula is the molecule's identity card. It helps chemists to know precisely what's inside each molecule.

Avogadro's Constant

Avogadro's constant is a fundamental number in chemistry. It approximates the number of particles (atoms, molecules, etc.) in one mole of a substance, typically given as \(6.022 \times 10^{23}\). This number allows chemists to easily convert between the number of molecules and moles, a necessary step for practical lab work and understanding chemical reactions.
In the case of aspartame, once you calculate the moles present in a given mass, using Avogadro's constant lets you find out how many molecules that amount represents. This concept bridges the gap between the microscopic world of molecules and the macroscopic amounts measured in the laboratory.

Unit Conversion

Unit conversion is an essential step in chemical calculations since measurements can be represented in various units. In the problem of aspartame, mass is initially given in milligrams (mg). To calculate moles accurately, it's crucial to first convert this mass into grams (g), because molar mass uses grams per mole (g/mol) as its unit.
To convert from mg to g, divide the mass in mg by 1000. For example, \(1.00 \, \text{mg} = 0.00100 \, \text{g}\). Understanding and applying unit conversion ensures that calculations are consistent and accurate, enabling you to compare results correctly or use them in subsequent calculations.

Atomic Mass

Atomic mass is a hefty concept in chemistry that involves the weighted average mass of an atom's isotopes, expressed in atomic mass units (amu). This average reflects both the mass and relative abundance of isotopes in a naturally occurring element. Typical values can be found on the periodic table.
For calculating the molar mass of a compound like aspartame, knowing the atomic masses of its component elements is essential:

• Carbon (C): 12.01 g/mol
• Hydrogen (H): 1.01 g/mol
• Nitrogen (N): 14.01 g/mol
• Oxygen (O): 16.00 g/mol

These values are used to calculate the molar mass by multiplying each element's atomic mass by its abundance in the molecule, giving an overall measure that allows for the number of moles to be determined from a given mass.